No, you can't do that way shortcut. I recommend to download the PS3 3.61 firmware update into your PC first.

Minimum is:
(1) The new HDD (#1)
(2) Existing PS3 HDD (#2)
(3) A PC/Laptop with sufficient disk space (#3)

° Format #1 with Fat32 (1st post / google search)
° Plug #1 as USB into PS3, and run System Settings -> Backup Utility -> Backup (this backup may take 30min to several hours, depending on your size)
° Unplug #1 from PS3, plug into PC and copy all (including hidden files/folders, if any)
° Swap physical HDD into PS3
° Format #2 as Fat32, and copy back all files from PC -> #2 (to act as USB now)
° If you downloaded the firmware, you can copy to the proper folder also.
° Boot up PS3 (with #1) and follow instruction. It will take some time. If you have the firmware in HDD, you can save a lot of time from downloading 100MB+
° Once booted up, System Settings -> Backup Utility -> Restore

If it is the SAME PS3, all contents should be 99% there.

I have a spare USB HDD (#4), so I can skip the copy part 

Forgot to tell you some steps (which is mentioned in 1st page there also)

° Before start backup, remember to sync trophies (need online) for all the accounts in the PS3
° After restore, sync again for all accounts.